Mathematics

Questions tagged [transcendence-degree]

In abstract algebra, the transcendence degree of a field extension $L/K$ is a certain rather coarse measure of the “size” of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of $L$ over $K$.

59 questions
24
votes
1 answer

Are all transcendental numbers a zero of a power series?

So I came across the concept of extending the notion of irrationality to higher degree polynomials. The base case of this is standard irrationality. That is, a number is irrational if it cannot be expressed as the ratio of two integers. This is…
Ozaner Hansha
  • 707
10
votes
2 answers

Showing $[K(x):K(\frac{x^5}{1+x})]=5$?

Let $K$ be a field and $x$ be trnacendental over $K$. Compute $[K(x):K(\frac{x^5}{1+x})]$. I've never came across questions like these. It's easy to see that this degree is at most $5$, since: $$x^5-\alpha(x+1)=0$$ ($\alpha$ being…
35T41
  • 3,701
6
votes
2 answers

Is the subextension of a purely transcendental extension purely transcendental over the base field?

Let $K/E/F$ be extension of fields, where $K/F$ is purely transcendental . It is generally not true that $K/E$ is purely transcendental. For example, take $F(x)/F(x^2)/F$ . I wonder what is the situation for $E/F$ . Specifically, is $E/F$ purely…
fantasie
  • 1,851
6
votes
1 answer

Algebraic independence of elementary symmetric polynomials

I am following the book Lectures on Algebra by Abhyankar, p. 638. Let $k$ be a field, $x_1,x_2,\ldots,x_n$ be independent variables, and define $$e_1=x_1+x_2+\cdots+x_n\\ e_2=\sum_{i<j} x_ix_j\\ \cdots\\ e_n=x_1x_2\cdots x_n.$$ Let…
Maths Rahul
  • 3,505
6
votes
2 answers

Show that $[k(t): k(t^4 + t) ] = 4$

Let $k$ be the field with $4$ elements, $t$ a transcendental over $k$, $F = k(t^4 + t)$ and $K = k(t).$ Show that $[K : F] = 4.$ I think I have to use the following theorem, but I'm not quite putting it together. If $P = P(t), Q = Q(t)$ are nonzero…
user525033
4
votes
1 answer

Proper subfields of $\mathbb{C}$ isomorphic to $\mathbb{C}$

It is known that $\mathbb{C}$ has proper subfields which are isomorphic to $\mathbb{C}$ , see this question; let $K$ be such subfield of $\mathbb{C}$ . Let $\iota: K \to \mathbb{C}$ , $\iota(k)=k$ for all $k \in K$ . Now, I am confused: (1) If $K…
user237522
  • 7,267
4
votes
0 answers

Concerning $k \subset L \subset k(x,y)$

The following is a known result in algebraic geometry: Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$ ). Let $L$ be a field such that $k \subset L \subset k(x,y)$ and $L$ is of transcendence degree two…
user237522
  • 7,267
4
votes
1 answer

Transcendence degree and Krull dimension of finitely generated algebras

Let $K$ be a field, and let $a_1,\dots,a_{n+1}$ be $n+1$ elements of a finitely generated $K$ -algebra $A$ of Krull dimension $n$ . Are the elements $a_1,\dots,a_{n+1}$ always algebraically dependent over $K$ ? I.e: Are the monomials…
Pierre-Yves Gaillard
  • 20,596
3
votes
2 answers

Is $\mathbb{R}(x,\sqrt{1+x^2})$ a purely transcendental extension of $\mathbb{R}$?

I'm working on Exercise 9, Section 19 of Morandi's Field and Galois Theory : If $K=\mathbb{R}(x,\sqrt{1+x^2})$ , show that there is a $t\in K$ with $K=\mathbb{R}(t)$ . Trying to deduce a workable $t$ , I assume if $K=\mathbb{R}(t)$ for some $t$ ,…
Adelaide Dokras
  • 1,723
3
votes
1 answer

$L/K$ is algebraic iff its transcendence degree is zero

On Wikipedia, I read that A field extension $L/K$ is algebraic if and only if its transcendence degree is $0$ . $[\Rightarrow]$ Suppose $L/K$ is an algebraic field extension, i.e., for every $l \in L$ , there exists a non-zero polynomial $g(X)$ in…
stoic-santiago
  • 15,089
3
votes
1 answer

Two algebraically closed fields are isomorphic if and only if they have the same transcendence degree over their prime fields.

Prove that two algebraically closed fields of the same characteristic are isomorphic if and only if they have the same transcendence degree over their prime fields. The prime field is isomorphism to $\mathbb{Q}$ or $\mathbb{F}_p.$ Denote the prime…
Yunhao
  • 545
  • 3
  • 10
3
votes
2 answers

Proof verification: transcendence degree additive in towers

I am trying to prove that if $k\subseteq E\subseteq F$ are field extensions, then $$\text{tr.deg}_k F=\text{tr.deg}_k E+\text{tr.deg}_E F.$$ If $A=\{a_1,\ldots, a_n\}$ is a transcendence basis for $E$ over $k$ and $B=\{b_1,\ldots, b_m\}$ is a…
SVG
  • 4,411
3
votes
1 answer

Can you determine the transcendence degree of an algebra by looking at a generating set?

Let $K$ be a field and $A$ be a $K$ -algebra generated (as $K$ -algebra) by a set $S$ . The transcendence degree of $A$ is $$ \operatorname{trdeg}(A) = \sup\{|T| : T \subset A,\, T \text{ algebraically independent}\} \quad . $$ Setting $$ N =…
Carlos Esparza
  • 3,817
3
votes
1 answer

Dimension of scheme of finite type over a field under base change (Hartshorne Ex. II.3.20)

Consider an integral scheme $X$ of finite type over a field $k$ . If $k\subseteq k'$ is a field extension, then the scheme $X' = X\otimes_k k'$ is not necessarily integral. For instance, take $X = \operatorname{Spec} \mathbb{R}[x,y]/(x^2+y^2)$ over…
Davide Cesare Veniani
  • 740
2
votes
0 answers

Schanuel's Conjecture $\implies $ no surprises on the integers. $2^t+3^t=1 \implies t\notin\overline{\mathbb{Q}}$

Simple Question: Is the proof given below correct? Let $t$ satisfy $2^t+3^t=1$ . We have $t \approx -0.787884911025869783628555917298434738269083137354182194199 \dots$ according to wolfram alpha. Then $t$ is demonstrably transcendental assuming…
Mason
  • 4,658
1
2 3 4